Free fall and elapsed time

The previous lesson showed that slower clocks lower down and objects falling downwards go together, and how strongly the two are linked. This lesson shows why. It needs one more fact about clocks: how much time a clock records depends on its motion as well as its height.

Taking a clock on a journey

A clock’s motion affects how much time it records. Imagine a spacecraft carrying a clock past a row of synchronised clocks floating at rest relative to one another. Compare their readings as it passes. The spacecraft’s clock accumulates less time than the row of clocks records for the same journey.

At everyday speeds, the difference is tiny. At about 86.6% of light speed, the travelling clock records one second for every two seconds measured by the row of clocks. The time recorded by a clock along its own journey is called proper time.

Two clocks between the same two events

Imagine throwing a ball straight up and catching it at the same height two seconds later. Attach a tiny clock to it, and keep another beside your hand.

Both clocks start together and meet again at the catch. Their readings differ slightly because one stayed at the same height while the other travelled up and back.

Height and speed both matter

The travelling clock spends some time higher up, where clocks run faster. It also moves relative to the ground, which reduces the time it records. Both effects contribute to the reading when it returns.

We can compare other journeys too. A clock could be carried a little higher, or follow a lower arc, provided it still returns to the same place at exactly the same moment. Those alternative journeys may need an engine or some other force to produce them.

For the short flight shown here, the freely falling arc records more time than nearby alternative paths. With no air resistance and gravity treated as constant, a launch speed of about 9.8 metres per second gives a two-second flight and a maximum height of about 4.9 metres.

Straight means the most clock time

In Force and motion, an object with no force acting on it carried on in a straight line. In How free-fall paths reveal curvature, the spacetime version of a straight line was called a geodesic. The clock comparison tells us how to recognise one.

Between two events, the straight path through spacetime is the one with the most time on a clock carried along it. Every nearby path between the same events records less. In the figure, each alternative arc needs a hand or an engine to produce it, and each returns with less time on its clock than the free arc.

This is Newton’s first law in spacetime form. An object left alone carries on as it was already going, and the path that results is the one with the most clock time.

Why the path bends downwards

The comparison so far fixed both ends of the journey. A falling ball does not need a destination, because the comparison also holds for every short piece of a free path. Take any two nearby moments on it, and the piece between them must have more clock time than any other route between the same two points.

That rules out a path that bends upwards. Picture a ball at rest that starts to rise on its own, slowly at first and then faster. A steady climb between the same two points is higher at every moment in between, where clocks run faster, and it never needs the fast finish that slows the rising ball’s clock. It has more clock time on both counts, so the upward bend cannot be a free path.

A downward bend is higher than a steady path between the same points, but it moves faster. Too little bend misses some of the height; too much brings too much speed. The two balance at exactly one amount of bend, and the faster the clock rate changes with height, the larger that amount is. That is the relationship between the clock-rate gradient and the acceleration from the previous lesson. Near Earth’s surface, it gives 9.8 metres per second squared.

So a ball released from rest goes down. Going up is allowed: a thrown ball rises, but it slows as it rises, so its path bends downwards the whole time. A ball at rest has no upward speed, and rising would mean bending upwards.

The ball does not need to choose

This can sound as though the ball selects the journey that gives its clock the most time. The comparison is something we calculate; the ball only follows the motion set by its initial velocity and the surrounding spacetime.

Describing the path through its clock reading gives the same answer as working out the motion step by step. It is the same free-fall rule, seen through the clocks.

How far the rule applies

The maximum-time result applies to a short enough section of a free-fall path, compared with nearby alternatives between the same events. It is not a guarantee that every long journey through curved spacetime gives the greatest possible clock reading.

The more general mathematical rule uses the word stationary. It means that the elapsed-time result has a locally flat slope as we vary the path, as a graph does at the top of a smooth hill. In this example, that point is a maximum.

Clock rates are enough to describe slow objects near Earth. Light moves fast enough that the geometry of space matters as well, and the next lesson looks at that.